46
5 CONCLUSIONS
In this paper we have considered the different approaches used in the parametrization of the
relationship of the main market risk factors of KGHM capital group. Based on the visual
inspection and the correlogram-like picture we have selected four market risk factors that are
mostly dependent. We have proposed three approaches to quantify the dependence structure
for analyzed data, namely classical Pearson, Kendall’s rank correlations and exponentially
weighted Pearson correlation coefficient. Based on the mentioned measures of dependence
determined as functions of time we have indicated their differences based on the real time
series analysis. The main result of this paper is the fact that the weighted correlation coefficient seems to be the most appropriate in the description of the structure of dependence
for analyzed market risk factors. Moreover, the correlation coefficients calculated for daily
data demonstrate proper dynamic of the dependence which corresponds with the behavior
of the data caused by market events. The presented approach can be a base for analyzing the
models which take under consideration the dependence of the data expressed in terms of
non-classical measures of dependence. One of the most important questions, which can be
addressed here is whether there are regime changes of relations between risk factors and if
they exists how it can be used for forecasting purposes.
REFERENCES
[1] P.J. Brockwell, R.A. Davis, Introduction to Time Series and Forecasting, Springer, 2002.
[2] W.W. Daniel, Kendall’s tau, Applied Nonparametric Statistics (2nd ed.). Boston: PWS-Kent.
365–377. 1990.
[3] M. Kendall, J.D. Gibbons, Rank Correlation Methods. Charles Griffin Book Series (5th ed.).
Oxford: Oxford University Press, 1990.
[4] D.G. Bonett, T.A. Wright, Sample size requirements for estimating Pearson, Kendall, and Spearman correlations, Psychometrika. 65 (1), 23–28, 2000.
[5] G. Samorodnitsky, M.S. Taqqu, Stable Non-Gaussian Random Processes, Chapman & Hall, 1994.
[6] D. Rosadi, M. Deistler, Estimating the codifference function of linear time series models with infinite variance, Metrika 73(3), 395–429, 2011.
[7] A. Wyłomańska, A. Chechkin, I.M. Sokolov, J. Gajda, Codifference as a practical tool to measure
interdependence, Physica A 421, 412–429, 2015.
[8] C.M. Gallagher, A method for fi tting stable autoregressive models using the autocovariation function, Statistics & Probability Letters 53, 381–390, 2001.
[9] P. Kruczek, A. Wyłomańska, M. Teuerle, J. Gajda, The modified Yule-Walker method for alphastable time series models, Physica A 469, 588–603, 2017.
[10] G. Żak, M. Teuerle, A. Wyłomańska Agnieszka, R. Zimroz, Measures of dependence for alphastable distributed processes and its application to diagnostics of local damage in presence of impulsive noise, Shock and Vibration, vol. 2017, Article ID 1963769, 9 pages, 2017.
[11] M. Shao, C.L. Nikias, Signal processing with fractional lower order moments: stable processes and
their applications, Proceedings of the IEEE 81(7), 986–1010, 1993.
[12] G. Żak, A. Wyłomańska, R. Zimroz, Periodically impulsive behaviour detection in noisy observation based on generalised fractional order dependency map, Applied Acoustics 144, 31–39, 2019.
[13] F. Pozzi, T. Di Matteo, T. Aste, Exponential smoothing weighted correlations, Eur. Phys. J. B 85,
175, 2012.
[14] A. Meucci, Risk and Asset Allocation, Springer, 2005.
[15] R. Litterman, K. Winkelmann, Estimating covariance matrices, in Goldman Sachs, Risk Management Series, 1998.
[16] D. Quade, I.A. Salama, A survey of weighted rank correlation, in Order Statistics and Nonparametrics: Theory and Applications, edited by P.K. Sen, I.A. Salama, Elsevier, 1992.
[17] J. Svensson, The asymptotic spectrum of the EWMA covariance estimator, Physica A 385, 621–
630, 2007.
[18] M.G. Kendall, A new measure of rank correlation, Biometrika 30, 81–93, 1938.
[19] M.G. Kendall, Rank Correlation Methods, Charles Griffin & Co Ltd., 1948.
5 CONCLUSIONS
In this paper we have considered the different approaches used in the parametrization of the
relationship of the main market risk factors of KGHM capital group. Based on the visual
inspection and the correlogram-like picture we have selected four market risk factors that are
mostly dependent. We have proposed three approaches to quantify the dependence structure
for analyzed data, namely classical Pearson, Kendall’s rank correlations and exponentially
weighted Pearson correlation coefficient. Based on the mentioned measures of dependence
determined as functions of time we have indicated their differences based on the real time
series analysis. The main result of this paper is the fact that the weighted correlation coefficient seems to be the most appropriate in the description of the structure of dependence
for analyzed market risk factors. Moreover, the correlation coefficients calculated for daily
data demonstrate proper dynamic of the dependence which corresponds with the behavior
of the data caused by market events. The presented approach can be a base for analyzing the
models which take under consideration the dependence of the data expressed in terms of
non-classical measures of dependence. One of the most important questions, which can be
addressed here is whether there are regime changes of relations between risk factors and if
they exists how it can be used for forecasting purposes.
REFERENCES
[1] P.J. Brockwell, R.A. Davis, Introduction to Time Series and Forecasting, Springer, 2002.
[2] W.W. Daniel, Kendall’s tau, Applied Nonparametric Statistics (2nd ed.). Boston: PWS-Kent.
365–377. 1990.
[3] M. Kendall, J.D. Gibbons, Rank Correlation Methods. Charles Griffin Book Series (5th ed.).
Oxford: Oxford University Press, 1990.
[4] D.G. Bonett, T.A. Wright, Sample size requirements for estimating Pearson, Kendall, and Spearman correlations, Psychometrika. 65 (1), 23–28, 2000.
[5] G. Samorodnitsky, M.S. Taqqu, Stable Non-Gaussian Random Processes, Chapman & Hall, 1994.
[6] D. Rosadi, M. Deistler, Estimating the codifference function of linear time series models with infinite variance, Metrika 73(3), 395–429, 2011.
[7] A. Wyłomańska, A. Chechkin, I.M. Sokolov, J. Gajda, Codifference as a practical tool to measure
interdependence, Physica A 421, 412–429, 2015.
[8] C.M. Gallagher, A method for fi tting stable autoregressive models using the autocovariation function, Statistics & Probability Letters 53, 381–390, 2001.
[9] P. Kruczek, A. Wyłomańska, M. Teuerle, J. Gajda, The modified Yule-Walker method for alphastable time series models, Physica A 469, 588–603, 2017.
[10] G. Żak, M. Teuerle, A. Wyłomańska Agnieszka, R. Zimroz, Measures of dependence for alphastable distributed processes and its application to diagnostics of local damage in presence of impulsive noise, Shock and Vibration, vol. 2017, Article ID 1963769, 9 pages, 2017.
[11] M. Shao, C.L. Nikias, Signal processing with fractional lower order moments: stable processes and
their applications, Proceedings of the IEEE 81(7), 986–1010, 1993.
[12] G. Żak, A. Wyłomańska, R. Zimroz, Periodically impulsive behaviour detection in noisy observation based on generalised fractional order dependency map, Applied Acoustics 144, 31–39, 2019.
[13] F. Pozzi, T. Di Matteo, T. Aste, Exponential smoothing weighted correlations, Eur. Phys. J. B 85,
175, 2012.
[14] A. Meucci, Risk and Asset Allocation, Springer, 2005.
[15] R. Litterman, K. Winkelmann, Estimating covariance matrices, in Goldman Sachs, Risk Management Series, 1998.
[16] D. Quade, I.A. Salama, A survey of weighted rank correlation, in Order Statistics and Nonparametrics: Theory and Applications, edited by P.K. Sen, I.A. Salama, Elsevier, 1992.
[17] J. Svensson, The asymptotic spectrum of the EWMA covariance estimator, Physica A 385, 621–
630, 2007.
[18] M.G. Kendall, A new measure of rank correlation, Biometrika 30, 81–93, 1938.
[19] M.G. Kendall, Rank Correlation Methods, Charles Griffin & Co Ltd., 1948.